8. THE VALUE OF

Assuming that the inferred motions are real and generated by GI, they
can be used to estimate .
Evidence from virialized systems on smaller scales suggest a low-density
universe of ~ 0.1-0.2,
but these values may be biased. The spatial variations of the
large-scale velocity field now allow measuring the mass density in a volume
closer to a ``fair'' sample. One family of methods is based on
comparing the dynamical fields derived from velocities to the fields
derived from galaxy redshifts (Section 8.1,
8.2). These methods can be
applied in the linear regime but they always rely on the assumed biasing
relation between galaxies and mass often parametrized by b, so they
provide an estimate of f() / b.
Another family of methods measures
from
redshift surveys alone, based on
z-space deviations from isotropy
(Section 8.3). Finally, there are
methods which rely on non-linear effects in the velocity data alone, and
provide estimates of
independent of b (Section 8.4,
8.5). The various estimates of
and are summarized in
Table 1 at the end of the section. Note that the
errors quoted
by different authors reflect different degrees of sophistication in the
error analysis, and are in many cases underestimates of the true uncertainty.

Equation (19) is best at estimating v(0), the linear velocity of the LG
in the CMB frame due to the gravitational acceleration g(0) exerted
by the mass fluctuations around it. A comparison with the LG velocity of
627 ± 22 km s-1 as given by the CMB dipole is
a direct measure of
. One expects
to obtain a lower bound on because v ~ 600 km s-1 is very
improbable in a low-density universe. One way to
estimate g(0) is from a whole-sky galaxy survey where only the
angular positions and the fluxes (or diameters) are observed,
exploiting the coincidence of nature that both the
apparent flux and the gravitational force vary as
r-2. If L M, then the vector sum of the fluxes in a volume-limited
sample is g(0)
due to the mass in that volume. This idea
can be modified to deal with a flux-limited sample once the luminosity
function is known, and applications to the combined UGC/ESO
diameter-limited catalog of optical galaxies yield opt
values in the range 0.3-0.5
(Lahav 1987;
Lynden-Bell et al. 1989).
These estimates suffer from limited sky coverage, uncertain corrections
for Galactic extinction, and different selection procedures defining the
north and south samples. The IRAS catalog provides a superior
sky coverage of 96% of the sky, with
negligible Galactic extinction and with fluxes observed by one telescope,
but with possible under-sampling of cluster cores
(Kaiser & Lahav 1989).
A typical estimate from the angular IRAS catalog is I = 0.9
± 0.2 (Yahil et al. 1986).

The redshift surveys provide the third dimension which could help in
deriving g(0) by Equation (19), subject to the
difficulties associated with discrete, flux-limited sampling
(Section 5). The
question is whether g(0) is indeed predominantly due to the mass
within the volume sampled, i.e., whether g(0) as computed from
successive concentric spheres converges interior to Rmax.
This is an issue of fundamental uncertainty
(e.g. Lahav et al. 1990;
Juszkiewicz et al. 1990;
Strauss et al. 1992b).
The r - z mapping (21) could either compress or rarify the
z-space volume elements depending on
the sign of u in the sense that an outflow makes the z-space
density z
smaller than the true density : z(x) (x) - 2[v(x) - v(0)]
·rhat / r. The varying
selection function adds to this geometrical effect [in analogy to n (r)
in the IM bias (Section 3.2)] and there is
contribution from dv / dr as well
(Kaiser 1987).
It is thus clear that the redshifts must be corrected to distances and
that any uncertainty in v(x) at large x or at
x = 0 would confuse the derived g(0). The latter is the Kaiser
``rocket effect'': if v(0) originates from a finite volume r <
r0 and the density outside r0 is uniform with
v = 0, then the measurements in z-space
introduce a fake g(0) in the direction of v(0) due to the matter
outside r0, and this g(0) is logarithmically
diverging with r. v(0) is uncertain because it is derived
like the rest of v(x) from the density distribution - not
from the CMB dipole. These difficulties in identifying convergence
limit the effectiveness of this method in determining
the PS on large scales and . The hopes for improvement by
increasing the depth are not high because the signal according to
conventional PS models drops with distance faster than the shot-noise.

Attempting to measure I from the IRAS data,
Strauss et al. (1992b)
computed the probability distribution of g(0) under several
models for the statistics of fluctuations,
via a self-consistent solution for the velocities and an ad hoc
fix to the rocket effect, which enabled partial corrections
for shot-noise, finite volume, and small-scale non-linear
effects. They confirmed that the direction of g(0)
converges to a direction only ~ 20° away from the CMB dipole, but
were unable to determine unambiguously whether |g(0)| converges
even within 100 h-1Mpc. A maximum likelihood fit and careful
error analysis constrained I to the range 0.4-0.85 with little sensitivity
to the PS assumed. Rowan-Robinson et al.
(1991,
1993) obtained
from the QDOT dipole
I
= 0.8+0.2-0.15.
Hudson's (1993b) best estimate
from the optical dipole is
opt = 0.72+0.37-0.18.

The volume-limited Abell/ACO catalog of clusters with redshifts
within 300 h-1Mpc was used to compute g(0) in a similar way
under the assumption that clusters trace mass linearly
(Scaramella et al.
1991). An apparent convergence was found by ~ 180 h-1Mpc
to the value g(0) 4860c km s-1. A comparison with the
LG-CMB motion of 600 km s-1 yields c 0.123, which
corresponds to opt 0.44 and I 0.56 if the ratios of biasing
factors are 4.5:1.3:1 (Section 7.1). A
similar analysis by
Plionis & Valdarnini (1991)
yielded convergence by ~ 150 h-1Mpc and
values
larger by ~ 30-80%.

The linear correlation found between mass density and galaxy density
(Section 6.2) can be used to estimate the
ratio . The
density v
determined by POTENT from velocities assuming = 1 relates in linear
theory to the true by
vf(),
while linear biasing assumes = b-1g, so
v = g.
Dekel et al. (1993) carried
out a careful
likelihood analysis using the POTENT mass density from the Mark II
velocity data and the density of IRAS 1.9 Jy galaxies, and found
I
= 1.3-0.6+0.75 at 95% confidence.
A similar analysis based on the Mark III and IRAS 1.2 Jy is in progress.
The degeneracy of and
b is broken in the quasi-linear regime,
where (v) is no
longer f-1. The compatible quasi-linear corrections in
POTENT and in the IRAS analysis allow a preliminary
attempt to separate these parameters, which yields for Mark II data
> 0.46 (95% level) if
bI > 0.5.
A correction for IM bias could reduce the 95% confidence limit
to > 0.3 at most. These
results are valid for linear biasing; possible non-linear biasing
may complicate the analysis because it is hard to distinguish
from non-linear gravitational effects.

The advantage of comparing densities is that they are local,
independent of reference frame, and can be reasonably corrected for
non-linear effects. The comparison can alternatively
be done between the observed velocities and those predicted from a
redshift survey, subject to limited knowledge of the quadrupole and
higher moments of the mass distribution outside the surveyed volume
and other biases.
Kaiser et al. (1991)
obtained from Mark II velocities versus QDOT
predictions I = 0.9+0.20-0.15. An
analysis by Roth (1994)
using IRAS 1.9 Jy galaxies yielded I = 0.6
± 0.3 (2).
Nusser & Davis (1994) implemented
a novel method based on
the Zel'dovich approximation in spherical harmonics to predict the
velocity dipole of distant shells from the IRAS 1.2 Jy redshift
survey and found in comparison to the dipoles derived from observed
velocities = 0.6 ± 0.2.

Similar comparisons with the optical galaxy fields indicate a similar
correlation between light and mass. A comparison at the velocity level
gives (Hudson 1994) opt = 0.5 ± 0.1, and a preliminary
comparison at the density level with 12 h-1Mpc smoothing indicates
(Hudson et al. 1994) opt 0.75 ± 0.2,
in general agreement
with the ratio of bopt/ bI 1.3-1.4 obtained by direct
comparison. Shaya et al.
(1994) applied the least-action
reconstruction method (Section 7.2) to a
redshift survey of several
hundred spirals within our local 30 h-1Mpc neighborhood and
crudely obtained by comparison to TF distances opt ~ 0.4.

Redshift samples, which contain hidden information about velocities, can
be used on their own to measure . The clustering, assumed
isotropic in real space, x, is anisotropic in z-space, z,
where z = r + xhat ·v displaces
galaxies along the preferred direction xhat. While virial
velocities on small scales stretch clusters into ``fingers of god''
along the line of sight, systematic infall
motions enhance large-scale structures by artificially squashing them
along the line of sight. The linear approximation -·v = g
indicates that the effect is -dependent because -·v is
related to the anisotropy in z-space while g is isotropic, so
the statistical deviations from isotropy can tell (e.g.
Sargent & Turner 1977).

Kaiser (1987) showed in linear
theory that the anisotropic Fourier PS in
z-space is related to the real-space PS of mass density, P
(k), via

Pz (k, µ)= P (k) (1 + µ2)2,
(25)

where µkhat ·xhat. This relation is valid only for a fixed
µ, i.e., in a distant volume of small solid angle
(Zaroubi & Hoffman 1994), but
there are ways to apply it more generally.
The redshift PS can be decomposed into Legendre polynomials,
Pl (µ),
with even multipole moments Plz
(k),

Pz(k, µ) = l=0Plz(k)Pl (µ),

Plz(k) = (2l +
1) / 2 -1+1 dµPz
(k, µ)Pl (µ).
(26)

Based on Equation (25) the first two non-vanishing moments are

P0z(k) = (1 + (2 / 3) + (1 / 5)2) P
(k),

P2z(k) = ((4 / 3) + (4 / 7)2) P
(k),
(27)

so the observable ratio of quadrupole to monopole
is a function of independent of P (k).
A preliminary application to the 1.2 Jy IRAS
survey yields I ~ 0.3-0.4 at wavelength 30-40 h-1Mpc,
suspected of being an underestimate because of non-linear
effects out to ~ 50 h-1Mpc
(Cole et al. 1993).
Peacock & Dodds (1994) developed a
method for reconstructing the linear
PS and they obtain I = 1.0 ± 0.2.

The distortions should be apparent in the z-space two-point correlation
function, z (rp, ), which is the excess of pairs with
separation
along the line of sight and rp transversely
(Davis & Peebles 1983).
The contours of equal , assumed round in r-space, appear in
z-space elongated along the line of sight at small separations
and squashed on large scales depending on .
Hamilton (1992;
1993) used the multiple moments of
z, in analogy to Equations (26-27),
and his various
estimates from the 1.9 Jy IRAS survey span the range
I
= 0.25-1.
Fisher et al. (1994a)
computed z(rp, )
from the 1.2 Jy IRAS survey, and derived the first two pair-velocity
moments. Their attempt to use the velocity dispersion via the Cosmic
Virial Theorem led to the conclusion that this is a bad method for estimating
, but the mean, <
v12 > = 109+64-47 at
10 h-1Mpc, yielded I =
0.45+0.27-0.18. The drawbacks of
using versus PS
are that (a) the uncertainty in the mean
density affects all scales in whereas it is limited to the k = 0
mode of the PS, (b) the errors on different scales in are
correlated whereas they are independent in a linear PS for a Gaussian
field, and (c) mixes different physical scales, complicating the
transition between the linear and non-linear regimes. Non-linear
effects tend to make all the above results underestimates.

A promising method that is tailored to deal with a realistic redshift
survey of a selection function (r) and does not rely on the subtleties of Equation (25)
is based on a weighted spherical harmonic
decomposition of z(z)
(Fisher et al. 1994b),

a lmz = d3z(r)f (z) [1 + z(z)] Ylm(zhat),

< |a lmz| 2 > = (2/) 0dk k2P
(k) | lr(k) + lc(k) | 2.
(28)

The arbitrary weighting function f (z) is vanishing at infinity
to eliminate surface terms. The mean-square of the harmonics is derived
in linear theory assuming that the survey is a ``fair'' sample,
and r and
c are explicit
integrals over r of certain expressions involving (r), f (r),
Bessel functions and their derivatives. The first term represents real
structure and the second is the correction embodying the z-space
distortions. The harmonic PS in z-space, averaged over m,
is thus determined by P (k) and , where the
z-space distortions appear as
a -dependent
excess at small l. The harmonic PS derived
from the 1.2 Jy IRAS survey yields I = 1.0
± 0.3 for an assumed
8 = 0.7
(motivated by the IRAS ,
Fisher et al. 1994a),
with an additional systematic uncertainty of ± 0.2 arising from the
unknown shape of the PS.

The methods for measuring from redshift distortions are
promising because they are relatively free of systematic errors and
because very large redshift surveys are achievable in the near future.
With a sufficiently large redshift survey, one can even hope to be able
to use the non-linear effects to determine and b separately.

Assuming that the initial fluctuations are a random Gaussian
field, the one-point PDF of smoothed density develops a characteristic
skewness due to non-linear effects early in the
quasi-linear regime (Section 7.2). The
skewness of is given in second-order
perturbation theory by < 3 > / < 2 >2 (34/7 - 3 - n),
where n is the effective power index near the
smoothing scale
(Bouchet et al.
1992). Since this ratio
for is practically
independent of ,
and since ·v ~ -f, the
corresponding ratio
for ·v
strongly depends on , and
in second-order
(Bernardeau et al. 1994).

T3
< (·v)3 > / < (·v)2 >2
-f()-1
(26 / 7 - 3 - n).
(29)

Using N-body simulations and 12 h-1Mpc smoothing one
indeed finds T3 = -1.8 ± 0.7 for = 1 and T3
= -4.1 ± 1.3 for = 0.3,
where the quoted error is the cosmic scatter for a sphere of radius
40 h-1Mpc in a CDM universe (H0 = 75,
b = 1). A preliminary estimate
of T3 in the current POTENT velocity field within 40
h-1Mpc is -1.1 ± 0.8, the
error representing distance errors. With the two errors added in
quadrature, = 0.3 is
rejected at the ~ 2 level
(somewhat sensitive to the assumed PS).

Since the PDF contains only part of the information stored in the data
and is in some cases not that sensitive to the IPDF
(Section 7.2), a
more powerful bound can be obtained by using the detailed
v(x) to recover the IPDF,
and use the latter to constrain . This is done by comparing the
-dependent IPDF recovered
from observed velocities to an
assumed IPDF
(Nusser & Dekel 1993), most
naturally a Gaussian as recovered from IRAS density
(Section 7.2). The velocity out of
POTENT Mark II within a conservatively selected volume was fed into the IPDF
recovery procedure with
either 1 or 0.3, and the errors due to
distance errors and cosmic scatter were estimated. The IPDF recovered
with = 1 is found
marginally consistent with Gaussian while the
one recovered with = 0.3
shows significant deviations. The
largest deviation bin by bin in the IPDF is ~ 2 for
= 1 and > 4 for = 0.3, and a similar rejection
is obtained with a 2-type statistic. The skewness and kurtosis
are poorly determined because of noisy tails but the replacements <
x|x| > and < |x| > allow a rejection of
= 0.3 at the (5-6) levels.

Figure 8. Maps of c inferred from the observed velocities near the
Sculptor void in the Supergalactic plane, for two values of .
The LG is marked by '+' and the void is confined by the Pavo part of
the GA (left) and the Aquarius extension of PP (right). Contour spacing
is 0.5, with c
= 0 heavy, c >
0 solid, and c
< 0 dotted. The heavy-dashed contours mark the illegitimate downward
deviation of c
below -1 in units of ,
starting from zero (i.e., c = -1), and decreasing with spacing -0.5.
The value = 0.2 is ruled
out at the 2.9 level
(Dekel & Rees 1994).

A diverging flow in an extended low-density region can provide a robust
dynamical lower bound on , based on the fact that large outflows
are not expected in a low- universe
(Dekel & Rees 1994). The
velocities are assumed to be induced by GI, but no assumptions need
to be made regarding galaxy biasing or the exact statistical nature
of the fluctuations. The derivatives of a diverging velocity field infer
a non-linear approximation to the mass density,
c(, ðv / ðx)
(Equation 6), which is an overestimate, c > , when the true
value of is
assumed. Analogously to 0 = -f()-1·v, the c inferred from a given diverging
velocity field becomes more negative when a smaller is assumed, and it may become
smaller than -1. The value of is bounded from below because mass
is never negative, -1.

The inferred c(x) smoothed at 12 h-1Mpc and
the associated
error field are derived by POTENT from
the observed radial velocities and, focusing on the deepest density
wells, the assumed is
lowered until c becomes significantly smaller than -1.
The most promising ``test case'' provided by the Mark III data seems to
be a broad diverging region centered near the supergalactic plane at the
vicinity of (X, Y) = (-25, -40) in h-1Mpc -
the ``Sculptor void'' of galaxies (Kauffman et al. 1991)
next to the ``Southern Wall'' (Figure 8). Values
of 1 are perfectly consistent
with the data, but c becomes smaller than -1 already for = 0.6.
The values = 0.3 and 0.2
are ruled out at the 2.4-, and
2.9 levels in terms of
the random error .
This is just a preliminary result. The systematic
errors have been partially corrected for in POTENT, but a
more specific investigation of the SG biases affecting the smoothed
velocity field in density wells is required.
For the method to be effective one needs to find a void that is (a)
bigger than the correlation length for its vicinity to represent the
universal , (b)
deep enough for the lower bound to be tight, (c)
nearby enough for the distance errors to be small, and (d) properly
sampled to trace the velocity field in its vicinity.